A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$

Question

I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\leq k\leq n-1)$: \begin{align*} x_{2k-1}&=\frac{a^2}{z_n}\cos \frac{z_k}{a}\\ x_{2k}&=\frac{a^2}{z_n}\sin \frac{z_k}{a}\\ x_{2n-1}&=a\int^{z_n}\frac{\sqrt{z_n^2-(n-1)a^2}}{z_n^2}dz_n \end{align*}

I'm trying to prove the following statements:

1. It is a local isometric immersion.

Here, taking $\phi:\mathbb H^n\to \mathbb R^{2n-1}$ given by $\phi(z_1,\dots,z_n)=(x_1,\dots,x_{2n-1})$ I imagine that $\phi^*g_{\mathbb R^{2n-1}}=g_{\mathbb H^n}$ which would prove that is a isometric immersion, but the conditions for $x_{2n-1}$ to be well defined make it only a local immersion.

2. It has a constant curvature $K\equiv -1/a^2$.

This is where I have some problems: is this a consequence of the above result? I'm trying with Christoffel's symbols.

3. Any ideas to prove that image $\phi(z_1,\dots,z_n)$ is not a complete surface?

I started to see this example as a coincidence but I was thinking a bit about what happens in $\mathbb R^3$: there are $3$ types of smooth surfaces of revolution with negative constant curvature given by $x(u,v)=(f(v)\cos u,f(v)\sin u,g(v))$, this is clear when solving $$K=-\frac{f''(v)}{f(v)}.$$ Is there something similar in $\mathbb R^{2n-1}$, how many surfaces with these characteristics exist? is there a differential equation as in $\mathbb R^3$?

Answer 1

The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.

Now, some additional things that you might be interested to:

1. There are many (explicit in some cases) local isometric immersions from $\mathbb H^n$ to $\mathbb R^{2n-1}$. These can be constructed by using either the Ribaucour or the Bäcklund transformation (for instance, see the papers by Dajczer-Tojeiro and Tenenblat-Terng).
2. Local isometric immersions of the hyperbolic plane $\mathbb H^2$ into $\mathbb R^3$ imply "local" solutions, that is, solutions that are not defined on the whole $\mathbb R^2$, of the sine-Gordon equation and vice versa. Therefore, it follows from Hilbert's theorem that there is no "global" solution, that is, a solution defined on the whole plane $\mathbb R^2$, of the sine-Gordon equation. Just like in the case of dimension two, the same also happens in the higher dimensional case where now you will end up with a system of PDES (see for instance Dajczer-Tojeiro). We can have local solutions to this system but we don't know if there exists any global. The existence of a global solution would imply the existence of a global isometric immersion of $\mathbb H^n$ into $\mathbb R^{2n-1}$, which would give a non affirmative answer to the major still open problem (in submanifolds) up to this day, which is the following conjectured extension of Hilbert's theorem:

There is no global isometric immersion from $\mathbb H^n$ to $\mathbb R^{2n-1}$

However, the above holds true in some very special cases. For instance:

• If the immersion is also minimal (the mean curvature vanishes) (see Moore).

I should also mention here that $\mathbb H^2$ admits no minimal immersion in any Euclidean space. (for a proof of this fact see either "Lectures on minimal submanifolds" by Lawson, or Bryant, or Di Scala).

• (weaker) If the immersion has also bounded mean curvature (see here)
• (even weaker) If also the length of the mean curvature of the immersion does not go to infinity too fast, that is, exponentially fast (see here)

I also recommend the following:

• The survey of Borisenko here
• Chapter 5 of the book of Dajczer and Tojeiro Submanifold Theory "Beyond an Introduction"